Constructing Histograms and Frequency Polygons

By the end of this lesson, learners should be able to

1. Calculate class midpoints and plot frequency polygons using straight line segments.
2. Anchor frequency polygons to the horizontal axis at imaginary preceding and succeeding class midpoints.
3. Convert discrete class limits into class boundaries to construct histograms for grouped data.

Key vocabulary

Frequency polygon

A line graph created by joining consecutive class midpoints and anchoring ends at zero frequency.

Joining the midpoints of adjacent classes with straight lines forms a closed frequency polygon.

Imaginary class

A hypothetical class before the first or after the last data class with zero frequency.

The anchor points lie at the midpoints of the imaginary class before and after the data.

Class midpoint

The central value of a class interval found by averaging the lower and upper limits.

The class midpoint for $25-34$ is $29.5$, plotted on the horizontal axis.

Histogram

A graphical display of grouped continuous data using adjacent rectangular bars with no spaces.

The mass distribution of learners is shown on a histogram with adjacent vertical bars.

Class boundaries

The true upper and lower limits that close gaps between adjacent discrete class intervals.

Class boundaries for $45-54$ are $44.5$ and $54.5$, marking bar edges.

Lesson preview

Slide 1 · Constructing Frequency Polygons and End-Anchoring RulesOpening · 12 min

A frequency polygon provides a clear linear representation of grouped data. To construct a frequency polygon, calculate the midpoint of each class interval using $\text{Midpoint}=\frac{\text{Lower Limit}+\text{Upper Limit}}{2}$. Plot each frequency against its midpoint on a Cartesian plane, then join adjacent points with straight line segments using a ruler. Finally, anchor the polygon by extending lines to the horizontal axis at the midpoints of imaginary classes immediately preceding the first class and following the last class at frequency zero.

• Calculate class midpoints by adding lower and upper class limits and dividing by two.
• Mark each class frequency vertically against its matching class midpoint along the horizontal axis.
• Connect adjacent plotted midpoint coordinates in sequence using straight line segments drawn with a ruler.
• Extend the polygon to touch the horizontal axis at midpoints of imaginary classes before and after the data.

Worked example. Calculate given class midpoints: $\frac{15+24}{2}=19.5$, $\frac{25+34}{2}=29.5$, and $\frac{35+44}{2}=39.5$. Determine the uniform class width $w=29.5-19.5=10$. Subtract $10$ from the first midpoint to anchor the start: $19.5-10=9.5$ at $f=0$. Add $10$ to the final midpoint to anchor the end: $39.5+10=49.5$ at $f=0$. Answer: The class midpoints are $19.5$, $29.5$, and $39.5$; the anchor midpoints on the horizontal axis are $9.5$ and $49.5$..

Check question: If the first class interval in a dataset is $16-25$ with a class width of $10$, what is the midpoint of the imaginary preceding class where the frequency polygon touches the horizontal axis?

1. $10.5$
2. $15.5$
3. $5.5$
4. $30.5$

Show answer

$10.5$. The midpoint of the first class interval, $16-25$, is $\frac{16+25}{2}=20.5$. To find the midpoint of the imaginary preceding class, subtract the class width of $10$ from the first midpoint: $20.5-10=10.5$. This point will be plotted at frequency zero.

Slide 2 · Calculating Midpoints and Plotting Frequency PolygonsModel · 14 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 15 lessons in this sub-strand

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All 16 lessons in Statistics I

1. 1Gathering Data from Practical SituationsCollect data from real-life sources
2. 2Frequency Distribution Tables for Ungrouped and Grouped DataDraw a frequency distribution table for grouped and ungrouped data
3. 3Drawing Grouped Frequency Distribution TablesDraw a frequency distribution table for grouped and ungrouped data
4. 4Constructing Frequency Distribution Tables for Grouped and Ungrouped DataDraw a frequency distribution table for grouped and ungrouped data
5. 5Calculating Averages and Middle Values of Raw DataDetermine mean, mode and median of grouped and ungrouped data
6. 6Determining Median of Data and Introduction to Grouped Data MeanDetermine mean, mode and median of grouped and ungrouped data
7. 7Determining Mean, Modal Class, and Median of Grouped DataDetermine mean, mode and median of grouped and ungrouped data
8. 8Determining Mean, Mode, and Median of Grouped and Ungrouped DataDetermine mean, mode and median of grouped and ungrouped data
9. 9Representing Data Using Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
10. 10Constructing Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
11. 11Advanced Data Representation: Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
12. 12Interpreting Data from Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
13. 13Interpreting Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
14. 14Interpreting Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
15. 15Data Collection, Organisation, and Informed Decision MakingPromote data collection, organisation and representation for informed decision making
16. 16Data Interpretation and Decision MakingPromote data collection, organisation and representation for informed decision making